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Comparison theorems for deformation functors via invariant theory

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Abstract

We compare deformations of algebras to deformations of schemes in the setting of invariant theory. Our results generalize comparison theorems of Schlessinger and the second author for projective schemes. We consider deformations (abstract and embedded) of a scheme X which is a good quotient of a quasi-affine scheme \(X^\prime \) by a linearly reductive group G and compare them to invariant deformations of an affine G-scheme containing \(X^\prime \) as an open invariant subset. The main theorems give conditions for when the comparison morphisms are smooth or isomorphisms.

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Acknowledgements

We would in particular like to thank Dmitry Timashev for patiently explaining aspects of invariant theory that led to the the correct setting for our results. We are grateful to Nathan Owen Ilten, Manfred Lehn, Benjamin Nill and Arne B. Sletsøe for helpful discussions and answering questions.

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Authors and Affiliations

  1. Matematisk institutt, University of Oslo, Postboks 1053 Blindern, 0316, Oslo, Norway

    Jan Arthur Christophersen

  2. Faculty of Technology, Art and Design, Oslo Metropolitan University, PB 4 St. Olavs plass, 0130, Oslo, Norway

    Jan O. Kleppe

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  1. Jan Arthur Christophersen
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  2. Jan O. Kleppe
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Correspondence to Jan O. Kleppe.

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Christophersen, J.A., Kleppe, J.O. Comparison theorems for deformation functors via invariant theory. Collect. Math. 70, 1–32 (2019). https://doi.org/10.1007/s13348-018-0232-z

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